EMISSIVITY OF TWO-PHASE COMBUSTION PRODUCTS IN A SOLID-PROPELLANT ROCKET ENGINE

Leonid A.Dombrovsky

In this article, we consider the effect of particle size, volume fraction, and
temperature on spectral and integral emissivity of an isothermal volume of
combustion products. Following Dombrovsky (1996), the main results are given for
the simplest geometrical model of a homogeneous plane-parallel layer of the medium.
This model has been employed in early papers by Dombrovsky and Ivenskikh (1973)
and Dombrovsky (1974, 1976). One can find a similar problem statement in papers
by Pearce (1978), Modest (1981), Popov (1984), Der and Nelson (1985), Kuzmin et
al. (1989), Lee (1989), Brewster (1989), and Thynell (1990), where infinite cylindrical
volume was considered side by side with the plane-parallel layer. Papers by Pearce
(1978), Der and Nelson (1985), and Kuzmin et al. (1989) are also concerned with the
thermal radiation of aluminized propellant combustion products. The simple
model of isotropic scattering was assumed by Pearce (1978). More accurate
calculations for anisotropic scattering were performed by Dombrovsky and
Ivenskikh (1973) and Der and Nelson (1985). The results by Dombrovsky and
Ivenskikh (1973) are more general, i.e., Der and Nelson (1985) performed
calculations at only one radius of particles, a = 2.5 μm, without taking into
account any spectral dependence of aluminum oxide optical constants. A
simplified approximate method of the spectral emissivity calculations for
combustion products of an aluminized propellant has been proposed by
Kuzmin et al. (1989). The papers mentioned do not give the complete notion
about the main thermal radiation regularities for two-phase combustion
products under consideration. Following the monographs by Dombrovsky
(1996) and Dombrovsky and Baillis (2010), a complete discussion of thermal
radiation from a cloud of molten aluminum oxide particles is presented in this
article.

Consider the thermal radiation of a layer containing monodisperse droplets of
aluminum oxide melt. Obviously, the spectral hemispherical emissivity of the layer of
thickness H depends on the particle radius a, temperature T, and mass of the
particles per unit area of the disperse layer surface ρH, where ρ is the mass
concentration of particles. Computational results for the spectral emissivity ελ at T
= 3000 K are presented in Fig. 1. Calculations were made by use of the DP1
analytical solution (see the article, “The simplest approximation of double spherical
harmonics”). Similar dependences for the values of T = 2500 K and 3500 K can be
found in the book by Dombrovsky (1996). The density of an aluminum oxide melt
was determined by use of linear approximation of the experimental temperature
dependence (Kirshenbaum, and Cahill, 1960; Mitin and Nagibin, 1970) [see Eq. (8)
from the article, “Near-infrared properties of droplets of aluminum oxide
melt”].

One can see that the emission spectrum depends on the particle radius. In any
case, the greatest emissivity takes place in the visible spectral range where the
diffraction parameter x = 2πa/λ is large and transport albedo of a single particle
ωtr = Qstr/Qtr is relatively small (see article, “Near-infrared properties of droplets of
aluminum oxide melt”). Strong oscillations of the curves ελ(λ) observed at the
wavelength λ ≈ a can also be explained by consideration of single particle
radiative properties. The results for integral hemispherical emissivity defined
as

The integral emissivity increases monotonically with the particle concentration.
But the value of ε has a limit, whose value depends substantially on the particle
radius (see red curves in Fig. 2). Concentrations at which the limiting value of ε is
reached are different for various radii. For instance, the layer with ρH > 20-30 g/m2
is quite thick at a = 0.5 μm, but the concentration over 50 g/m2 is necessary to reach this
limit at a = 0.2 μm or 4 μm. The use of the concept of the limiting particle
concentration, for which the additional particles do not result in any variation of
the disperse layer emission, enables us to introduce the thickness of the
effective radiating layer H*, which depends on ρ and a. To understand the
physical sense of the limiting curve in Fig. 2, one should consider the spectral
dependences of the particle optical properties. With the variation of the
particle radius, the different regions of particle properties’ dependences on the
diffraction parameters correspond to the wavelength of the blackbody radiation
maximum,

(2)

Equation (2) is known as Wien’s displacement low, becauase it was developed
independently by Wilhelm Wien (1894) before the publication of Plank’s emissive
power law (Modest, 2003). For very small particles (a < 0.1 μm), the Rayleigh
scattering takes place at λ = λm and the scattering coefficient is small in comparison
to the absorption one. Therefore, the optically thick layer of such particles should be
almost absolutely black. With increasing the particle radius, the maximum of
transport scattering coefficient occurs near λm. For this reason, even the optically
thick layer of such particles has a low integral emissivity. The role of scattering
decreases with a further increase of the particle radius. As a result, the radiation of
an optically thick layer increases with the further increase of the particle size. It
should be noted that the limiting emissivity of the optically thick disperse
system is not so sensitive to the particle transport albedo ωtr. This fact is
quite obvious from the formula of the DP0 approximation (see the articles,
“The simplest approximation of double spherical harmonics” and “Two-flux
approximation”),

(3)

At ωtr << 1, this equation reduces to

(4)

This means that comparatively low values of ε∞ for a cloud of molten alumina
particles are explained by relatively high scattering of radiation by the particles. At
low particle concentrations, the character of dependence ε(a) is determined by the
absorption coefficient variation, since the role of scattering is negligible at small
optical thickness of the particle layer (see blue curves in Fig. 2). The maximum
of ratio Qa(x)/x at diffraction parameter x ≈ 6 leads to appearance of
the corresponding maximum of ε(a) located near a = 1 μm at ρH > 10
g/m2.

The integral emissivity calculations for a monodisperse layer were performed also
by assumption of isotropic scattering (see dashed-line curves in Fig. 2). This enables
us to estimate the role of scattering anisotropy. At low optical thickness, the
scattering does not influence the value of ε. If the layer has a large optical
thickness, the anisotropy of scattering leads to a considerable increase in the
integral radiation of the disperse system. In this case, the isotropic scattering
model yields an integral emissivity about 25% lower than that in the exact
solution.

The shape of a radiating volume (particle cloud) affects the emissivity when the
particle cloud is not optically thick. This effect can be easily estimated by comparison
of the analytical solutions to the following boundary-value problems (see the articles,
“Differential approximations,” “Two-flux approximation, and “P1 approximation of
spherical harmonics method”):

(5a)

(5b)

for n = 0 (plane-parallel layer), n = 1 (infinite cylinder), and n = 2 (sphere). These
solutions can be written in the form (Dombrovsky and Baillis, 2010)

(6)

where

(7)

Here, I0 and I1 are the modified Bessel functions (Abramowitz and Stegun, 1965). In
the limit of large optical thickness, we have f →1, and the above solutions are
indistinguishable. In the opposite case of optically thin cloud, the limiting expressions
for ελ are as follows:

(8)

A typical difference between integral emissivity of a plane-parallel layer and a
cylindrical volume containing monodisperse alumina particles is shown in Fig.
3.

The limiting value of integral hemispherical emissivity of optically thick volumes
containing the particles of an aluminum oxide melt is often used in engineering
calculations of radiative heat transfer in combustion chambers. This value depends on
the particle temperature and size distribution. The dependences ε∞(a) for
monodisperse particles are shown in Fig. 4. These computational results for particles
of radius a > 0.5 μm can be approximated with high accuracy by the simple
function

The radiation calculations for polydisperse systems are much more complicated
than those for monodisperse systems. Therefore, it is of interest to analyze a
possibility of changing the polydisperse layer to a monodisperse one in the
calculations. The problem of determination of the equivalent mean particle radius a*,
for which the integral emissivity of the monodisperse layer at a = a* is approximately
equal to that of the polydisperse one, has been considered by Dombrovsky (1976). It
was assumed that an equivalent monodisperse layer has the same mass concentration
of particles as that in the real medium. The calculations for large optical thickness
showed the following value:

(10)

In the case of a two-parameter gamma-distribution (10) from the article “Radiative
properties of polydisperse systems of independent particles,” Eq. (10) can be written
as

(11)

By using Eqs. (9)-( 11), one can find the integral emissivity ε∞ with an error of <2%
in the temperature range 2600 ≤ T ≤ 3600 K and typical size distributions (see Table
1). The emission spectrum of a polydisperse layer is also similar to the smoothed
spectrum of the monodisperse with a = a* even for wide size distributions
(Dombrovsky, 1976, 1996).

For optically thin disperse systems containing molten alumina particles, one can
use the value of a* = a43. But the monodisperse approach appears to be not
applicable for intermediate optical thicknesses in the vicinity of the maximum of ε(a)
(Dombrovsky, 1976).

A computational analysis of thermal radiation in typical combustion
chambers showed that it is not important to take into account the absorption
coefficient of gases by determination of the integral emissivity of the two-phase
combustion products of aluminized solid propellants. For example, the calculated
integral emissivity of an optically thick volume of combustion products at
temperature 3460 K, pressure 5M Pa, mass fraction of condensed phase 0.34, and
mole fractions of H2O-6.8%, CO2-0.6%, CO-25.9% is only 0.4% greater
than the emissivity of the condensed phase without gases (Dombrovsky,
1996).

The combustion products of aluminized propellants may contain some optically
inhomogeneous particles such as alumina particles with a core of aluminum,
solidifying particles of alumina, alumina particles covered by a thin layer of
soot, and even hollow solid alumina particles. The optical properties of these
particles and their effect on the thermal radiation of combustion products have
been analyzed in the book by Dombrovsky (1996) and are not reproduced
here.

The qualitative results of the above computational analysis of thermal radiation
of isothermal clouds of alumina particles may be of interest not only for heat transfer
problems in solid rocket engines or in special power plants. These results are also
applicable in general terms for disperse systems containing Mie particles of other
weakly absorbing substances. The monodisperse approximation with appropriate
choice of the equivalent particle size is also general enough. The latter will be
illustrated for quite different disperse systems considered in another section of this
chapter.

It should be noted that radiation heat transfer in a combustion chamber of a
solid-propellant rocket engine is not a simple problem described by
the only characteristic - the emissivity of an isothermal cloud of alumina particles.
Particularly, the radiation flux to the burning surface of aluminized solid
propellant as well as the radiation flux to the combustion chamber wall cannot be
determined without in-depth analysis of conjugated combined heat transfer
processes.